Approximate determination of an Origin—Earthquake-hunting in Japan—Determinations by direction of motion—Direction indicated by destruction of buildings—Direction determined by rotation—Cause of rotation—The use of time observations—Errors in such observations—Origin determined by the method of straight lines—The method of circles, the method of hyperbolas, the method of co-ordinates—Haughton’s method—Difference in time between sound, earth, and water waves—Method of Seebach.

One of the most practical problems which can be suggested to the seismologist is the determination of the district or districts in any given country from which earthquake disturbances originate. With a map of a country before us, shaded with tints of different intensity to indicate the relative frequency of seismic disturbance in various districts, we at once see the localities where we might dwell with the least disturbance, and those we should seek if we wish to make observational seismology a study. Before erecting observatories for the systematic investigation of earthquakes in a country, it would be necessary for us, in some way or other, to examine the proposed country to find out the most suitable district. The special problem of determining approximately the origin or origins of a set of earthquakes would be given to us. Having made this preliminary investigation, the next point is, by means of observatories so arranged that they could always work in conjunction with each other, to determine the origins more accurately. By knowing the origin from which a set of shocks spring we know the general direction in which we may expect the most violent disturbances, and we can arrange our seismometers accordingly.

Approximate determination of origins.—In 1880 I obtained a tolerably fair idea of the distribution of seismic energy throughout Japan, by compiling the facts obtained from some hundreds of communications received from various parts of the country respecting the number of earthquakes that had been felt.

The communications were replies to letters sent to various residents in the country and to a large number of public officers. By taking these records, in conjunction with the records made by instruments, it was ascertained that in Japan alone there were certainly 1,200 shocks felt during the year, that is to say, three or four shocks per day. The greater number of these shocks were felt along the eastern coast, commencing at Tokio, in the south, and going northwards to the end of the main island. These shocks were seldom felt on the west coast. It appeared as if the central range of mountains formed a barrier to their progress. Similarly, ranges of mountains to the south-west of Tokio prevented the shocks from travelling southwards. Proceeding in this way the conclusion was arrived at that the west coast, the southern part of Japan, and the islands of Shikoku and Kiushiu, had their own local earthquakes.

Earthquake-hunting.—These preliminary enquiries having shown that the northern part of Japan was a better district for seismological observations than the southern half, the next step was to subject the northern half to a closer analysis. This analysis was commenced by sending to all the important towns, from thirty to one hundred miles distant from Tokio, bundles of postcards. These were entrusted to the local government offices with a request that each week one of these cards would be returned to Tokio stating the number of shocks felt. In this way it was quickly discovered that the majority of shakings emanated from the north and east, and seldom, if ever, passed a heavy range of mountains to the south. The barricade of postcards was then extended farther northwards, with the result of surrounding the origin of certain shocks amongst the mountains, whilst others were traced to the sea shore. By systematically pursuing earthquakes it was seen that many shocks had their origin beneath the sea—they shook all the places on the north-east coast, but it was seldom that they crossed through the mountains, forming the backbone of the island, to disturb the places on the west coast.

The actual results obtained in three months by this method of working are shown in the accompanying map, which embraces the northern half of the main island of Nipon and part of Yezo. The shaded portion of the map indicates the mountainous districts, which are traversed by ranges varying in height from about 2,000 and 7,000 feet. The dotted lines show the boundaries of the more important groups of earthquakes which were recorded.

I. is the western boundary of earthquakes, which at places to the eastward are usually felt somewhat severely. Some of these have been felt the most severely at or near Hakodadi, whilst farther south their effects have been weak. Occasionally the greatest effect has been near to Kameishi. Sometimes these earthquakes terminate along the western boundaries of III. or IV., not being able to pass the high range of mountains which separate the plain of Musashi from Kofu.

II. is the boundary of a shock confined to the plain which surrounds Kofu. These earthquakes are evidently quite local. Many of the disturbances have evidently originated beneath the ocean, having come in upon the land in the direction of the arrows a or b.

III. This line indicates the boundary of a group of shocks which are often experienced in Tokio. These may come in the directions d, e, or f. It is probable that some of them originate to the eastward of Yokohama, on or near to the opposite peninsula.

IV. V. and VI. The earthquakes bounded by these lines probably originate in the directions c or d.

VII. The earthquakes bounded by this line probably come from the direction e.

VIII. This line gives us the boundary of earthquakes which may come from the direction b.

The above boundaries sometimes do not extend so far to the westward as they are shown. At other times, groups like V. and VI. extend farther to the south-west. These earthquake boundaries, which so clearly show the effects of high mountains in preventing the extension of motion, have been drawn up, not from single earthquakes, but from a large series of earthquakes which have been plotted upon blank maps, and are now bound together to form an atlas. To give an idea of the material upon which I have been working, I may state that between March 1 and March 10, 1882, I received records of no less than thirty-four distinct shocks felt in districts between Hakodate and Tokio, and for each of these it is quite possible to draw a map. In addition to the boundaries of disturbances given in the accompanying map, other boundaries might be drawn for shocks which were more local in their character. The groups which contained the greatest number of shocks are III., IV., V., VI., and VII. By work of this description it was found that a very important group of earthquakes might be studied by a line of stations commencing at Saporo in the north, passing through Hakodate, down the east coast of the main island, to Tokio or Yokohama in the south. A further aid to the study of this group, together with the study of an important local group, might be effected with the help of a few additional stations properly distributed on the plain of Musashi, which surrounds Tokio. With this example before us it will be recognised that the choice of sites for a connected set of seismological observatories will often be more or less a special problem. If earthquake stations were to be placed in different directions around Tokio without preliminary investigation, it is quite possible that some of them might be so situated that they would seldom if ever work in conjunction with the remaining observatories, and therefore be of but little value. And this remark must equally apply to districts in other portions of the globe. The method is crude, and, so far as actual earthquake origins are concerned, it only yields results which are approximate. The crudeness and the want of absoluteness in the results is, however, more than counterbalanced by the certainty with which we are enabled to express ourselves with regard to such results as are obtained. Even when working with the best instruments we have at our command, unless we are employing some elaborate system, this method of working gives a most valuable check upon our instrumental records, and enables us to interpret them with greater confidence.

Determination of earthquake origins from the direction of motion.—If we assume that an earthquake is propagated from a centre as a series of waves, in which normal vibrations are conspicuous, and obtain at two localities, not in the same straight line with the origin, and sufficiently far separated from each other, the direction of movement of these normal motions, by drawing lines parallel to these directions through our two stations, the lines would intersect at a point above the required origin. If instead of two points we had three, or, better still, a large number, the results we should obtain ought to be still more certain. Unfortunately, it seems that earthquakes seldom originate from a given point, and, further, normal motions are not always (sufficiently) prominent. Sometimes, as has already been shown in the chapters on earthquake motion, they may be non-existent. It is probable, however, that difficulties of this sort are more usually associated with non-destructive earthquakes. Mallet regards the destructive effects of an earthquake as almost solely due to normal motions. If this be true, for destructive earthquakes, the problem is shorn of many of its difficulties. In cases where normal vibrations are not prominent, where we have only transverse vibrations, motions due to the interference of normal or transverse motions, or directions of motions due to the topographical or geological nature through which the disturbance has passed, the determination of the origin of an earthquake by observations on the direction in which the ground has been moving appears to be a problem which is practically without a solution. We will, therefore, only consider the determination of the origin of those earthquakes which have predominating directions in their movements, which directions we will consider as normal ones. The question which is, then, before us, is the determination of the direction of these normal movements. First of all we may take the evidence of our senses. In exceptional cases these have given results which closely approximate to the truth, but in the majority of cases such results are not to be relied upon, as the inhabitants of a town will, for the same shock, give directions corresponding to all points of the compass. Much, no doubt, depends upon the situation of the observer, and much, perhaps, upon his temperament. If he is sitting in a room alone, and is accustomed to making observations on an earthquake, on feeling the earthquake, if he concentrates his attention on the direction in which he is being moved, his observations may be of value. If, however, he is not so situated, and his attention is not thus concentrated, his opinions, unless the motion has been very decided in its character, are usually of but little worth.

Direction determined from destruction of buildings.—When an observer first sees a town that has been partially shattered by an earthquake, all appears to be confusion, and it is difficult to imagine that in such apparent chaos we are able to discover laws. If, however, we take a general view of this destruction and compare together similarly built buildings, it is possible to discover that similar and similarly situated structures have suffered in a similar manner. By carefully analysing the destruction we are enabled to infer the direction in which the destroying forces have acted. It was chiefly by observing the cracks in buildings, and the direction in which bodies were overthrown or projected, that Mallet determined the origin of the Neapolitan earthquake. From the observations given in Chapter VII. it would appear that, with destructive earthquakes, walls which are transverse to the direction of motion are most likely to be overturned, whilst, with small earthquakes, these walls are the least liable to be fractured.

From a critical examination of the general nature of the damage done on the buildings of a town, earthquake observers have shown that the direction of a shock may often be approximately determined. The direction in which a body having a regular form like a prismatic gravestone or a cylindrical column is overturned sometimes gives the means by which we can determine the direction from which a movement came.

The rotation of bodies.—It has often been observed that almost all large earthquakes have caused objects like tombstones, obelisks, chimneys, &c., to rotate.

One of the most natural and at the same time most simple explanations is to suppose that during the shock there had been a twisting, or backward and forward screw-like motion in the ground. Amongst the Italians and the Mexicans earthquakes producing an effect like this are spoken of as ‘vorticosi.’ In the Calabrian earthquake, not only were bodies like obelisks twisted on their bases, but straight rows of trees seem to have been left in interrupted zigzags. These latter phenomena have been explained upon the assumption of the interference of direct waves and reflected waves, the consequence of which being that points in close proximity might be caused to move in opposite directions. Reflections such as these would be most likely to occur near to the junction of strata of different elasticity, and it may be remarked that it is often near such places that much twisting has been observed.

Another way in which it is possible for twisting to have taken place would be by the interference of the normal and transverse waves which probably always exist in an earthquake shock, or by the meeting of the parts of the normal wave itself, one having travelled in a direct line from the origin, whilst the other, travelling through variable material, has had its direction changed.

Mallet, however, has shown that the rotation may have been in many cases brought about without the supposition of any actual twisting motion of the earth—a simple backward and forward motion being quite sufficient. If one block of stone rests upon another, and the two are shaken backwards and forwards in a straight line, and if the vertical through the centre of gravity of the upper block does not coincide with the point where there is the greatest friction between the blocks, rotation must take place. If the vertical through the centre of gravity falls on one side of the centre of friction, the rotation would be in one direction, whilst, if on the other side, the rotation would be in the opposite direction.

Although the above explanation is simple, and also in many cases probably true, it hardly appears sufficient to account for all the phenomena which have been observed.

Thus, for instance, if the stones in the Yokohama cemetery, at the time of the earthquake of 1880, had been twisted in consequence of the cause suggested by Mallet, we should most certainly have found that some stones had turned in one direction whilst others had been twisted in another. By a careful examination of the rotated stones, I found that every stone—the stones being in parallel lines—had revolved in the same direction, namely in a direction opposite to that of the hands of a watch.

As it would seem highly improbable that the centre of greatest friction in all these stones of different sizes and shapes should have been at the same side of their centres of gravity, an effect like this could only be explained by the conjoint action of two successive shocks, the direction of one being transverse to the other.

Although fully recognising the sufficiency of two transverse shocks to produce the effects which have been observed in Yokohama, I will offer what appears to me to be the true explanation of this phenomenon: it was first suggested by my colleague, Mr. Gray, and appears to be simpler than any with which I am acquainted.

If any columnar-like object, for example a prism which the basal section is represented by a b c d (see fig. 30), receives a shock at right angles to b c, there will be a tendency for the inertia of the body to cause it to overturn on the edge b c. If the shock were at right angles to d c, the tendency would be to overturn on the edge d c. If the shock were in the direction of the diagonal c a, the tendency would be to overturn on the point c. Let us, however, now suppose the impulse to be in some direction like e g, where g is the centre of gravity of the body. For simplicity we may imagine the overturning effect to be an impulse given through g in an opposite direction—that is, in the direction g e. This force will tend to tip or make the body bear heavily on c, and at the same time to whirl round c as an axis, the direction of turn being in the direction of the hands of a watch. If, however, the direction of impulse had been e' g, then, although the turning would still have been round c, the direction would have been opposite to that of the hands of a watch.

To put these statements in another form, imagine g e' to be resolved into two components, one of them along g c and the other at right angles, g f. Here the component of the direction g c tends to make the body tip on c, whilst the other component along g f causes revolution. Similarly g e may be resolved into its two components g c and g f', the latter being the one tending to cause revolution.

From this we see that if a body has a rectangular section, so long as it is acted upon by a shock which is parallel to its sides or to its diagonals, there ought not to be any revolution. If we divide our section a b c d up into eight divisions by lines through these directions, we shall see that any shock the direction of which passes through any of the octants which are shaded will cause a positive revolution in the body—that is to say, a revolution corresponding in its direction to that of the movements of the hands of a watch; whilst if its direction passes through any of the remaining octants the revolution will be negative, or opposite to that of the hands of a watch. From the direction in which any given stone has turned, we can therefore give two sets of limits between one of which the shock must have come.

Further, it will be observed that the tendency of the turning is to bring a stone, like the one we are discussing, broadside on to the shock; therefore, if a stone with a rectangular cross section has turned sufficiently, the direction of a shock will be parallel to one of its faces, but if it has not turned sufficiently it will be more nearly parallel to its faces in their new position than it was to its faces when in their original position.

If a stone receives a shock nearly parallel with its diagonal, on account of its instability it may turn either positively or negatively according as the friction on its base or some irregularity of surface bearing most influence. Similarly, if a stone receives a shock parallel to one of its faces, the twisting may be either positive or negative, but the probability is that it would only turn slightly; whereas in the former case, where the shock was nearly parallel to a diagonal, the turning would probably be great.

Determination of direction from instruments.—When speaking about earthquakes it was shown, as the result of many observations, that the same earthquake in the space of a few seconds, although it may sometimes have only one direction of motion, very often has many directions of motion. In certain cases, therefore, our records, if we assume the most permanent motions to be normal ones, give definite and valuable results. In other cases it is necessary to carefully analyse the records, comparing those taken at one station with those taken at another.

One remarkable fact which has been pointed out in reference to artificial earthquakes produced by exploding charges of gunpowder or dynamite, and also with regard to certain earthquakes, is that the greatest motion of the ground is inwards, towards the point from which the disturbance originated. Should this prove the rule, it gives a means of determining, not only the direction of earthquake, but the side from which it came.

Determination of earthquake origins by time observations.—The times at which an earthquake was felt at a number of stations are among the most important observations which can be made for the determination of an earthquake origin. The methods of making time observations, and the difficulties which have to be overcome, have already been described. When determining the direction from which a shock has originated, or determining the origin of the shock by means of time observations, it has been usual to assume that the velocity of propagation of the shock has been uniform from the origin. The errors involved in this assumption appear to as follows:—

1. We know from observations on artificial earthquakes that the velocity of propagation is greater between stations near to the origin of the shock than it is between more remote stations; and also the velocity of propagation varies with the initial force which produced the disturbance. If our points of observation are sufficiently close together as compared with their distance from the origin of the disturbance, it is probable that errors of this description are small and will not make material differences in the general results.

2. We have reasons for believing that the transit velocity of an earthquake is dependent on the nature of the rocks through which it is propagated. Errors which arise from causes of this description will, however, be practically eliminated if our observation points are situated on an area sufficiently large, so that the distribution of the causes tending to alter the velocity of a shock balance each other. It must be remarked, that causes of this description may also produce an alteration in the direction of our shock.

Other errors which may sometimes enter into our results, when determining the origin of shocks by means of observations on velocities, are the assumptions that the disturbance has travelled along the surface from the epicentrum and not in a direct line from the centrum. Again, it is assumed that the origin is a point, whereas it may possibly be a cavity or a fissure. Lastly, if we desire extreme accuracy, we must make due allowance for the sphericity of the earth and the differences of elevation of the observing stations.

I. The method of straight lines.—Given a number of pairs of points a₀, a₁, b₀, b₁, c₀, c₁, &c., at each of which the shock was felt simultaneously, to determine the origin.

Theoretically if we bisect the line which joins a₀ and a₁ by a line at right angles to a₀, a₁, and similarly bisect the lines b₀, b₁, c₀, c₁, all these bisecting lines a₀, a₁, b₀, b₁, c₀, c₁, &c., ought to intersect in a point, which point will be the epicentrum or the point above the origin.

This method will fail, first, if a₀, a₁, b₀, b₀, c₀, c₁ form a continuous straight line, or if they form a series of parallel lines.

Hopkins gives a method based on a principle similar to the one which is here employed—namely, given that a shock arrives simultaneously at three points to determine, the centre. In this case, the relative positions of the three points, where the time of arrival was simultaneous, must be accurately known, and these three points must not lie in a straight line, or the method will fail. For practical application the problem must be restricted to the case of three points which do not lie nearly in the same straight line.

II. The method of circles.—Given the times t₀, t₁, t₂, &c., at which a shock arrived at a number of places a₀, a₁, a₂, &c., to determine the position from which the shock originated.

Suppose a₀ to be the place which the shock reached first, and that it reached a₁, a₂, a₃, &c., successively afterwards.

Let t₁ - t₀ = a
t₂ - t₀ = b
t₃ - t₀ = c, &c.

With a₁, a₂, a₃, &c. as centres, describe circles with radii proportional to the known qualities a, b, c, &c., and also a circle which passes through a₀ and touches these circles. The centre of the last circle will be the epicentrum. The radii proportional to a, b, c, &c. may be represented by the quantities ax, bx, cx, &c., where x is the velocity of propagation of the shock.

It will be observed that the times at which the shock arrived at three places might alone be sufficient. If, instead of taking the times of arrival of a shock, the arrival of a sea wave be taken, the result would be a closer approximate to the absolute truth.

It will be observed that this method is not a direct one, but is one of trial. If, however, an imaginary case be taken, and three given points of observation, a₀, a₁, a₂, be plotted on a piece of paper, it will be found that it is not a difficult matter to determine two numbers proportional to a and b which will allow you to draw two circles so that they may be touched by a third circle drawn through a₀. This problem has practically been applied in the case of the arrival of a sea wave at a number of places on the South American coast, at the time of the earthquake of May 9, 1877. This is illustrated as follows. The places which were chosen were Huanillos, Tocopilla, Cobija, Iquique, Mejillones.

In the following table the first column gives the times at which the sea wave arrived at each of these places in Iquique time; in the second column the difference between these times and the time at which it reached Huanillos is given; in the third column the distances through which a sea wave, propagated at the rate of 350 feet per second, could travel during the intervals noted in the second column is given.

	Arrival of sea wave		Time after arrival at Huanillos	Distance at 350 feet per second
	h.	m.	minutes	miles
Huanillos	8	30	0	0
Tocopilla	8	32	2	8
Cobija	8	38	8	32
Iquique	8	40	10	40
Mejillones	8	46	16	64

The distances marked in the third column are used as radii of the circles drawn round the places to which they respectively refer.

The centre of the circle drawn to touch the circles of the first column, and at the same time to pass through Huanillos, is marked c.

The position from which the shock originated appears therefore to have occurred very near to a place lying in Long. 7° 15' W. and Lat. 21° 22' S.

The actual operations which were gone through in making the accompanying map were as follows. First, the places with which we had to deal were represented on a map in orthographical projection, the centre of projection being near to the centre of the map. This was done so that the measurements which were made upon the map might be more correct than those we should obtain from an ordinary chart where this portion of the world was not the centre of projection. Next, a number was taken as equal to the velocity with which the sea wave had travelled. The first velocity taken was about 400 feet per second—this being about the velocity with which, theoretically, it must have travelled in an ocean having a depth equal to that indicated upon the charts—also it seemed to have travelled at this rate from the various times of arrival as recorded at places along the coast. Circles were then drawn round Tocopilla, Cobija, Iquique, and Mejillones with radii equal to 2, 8, 10, and 15, each multiplied by (60 × 400). It was then seen by trial that it was impossible to draw a single circle which should touch four circles and also pass through Huanillos. These four circles were, in fact, too large. Four new but smaller circles, which are shown in the map, were next drawn, their radii being respectively equal to the numbers 2, 8, 10, and 16, each multiplied by (60 × 350), and it was found that a circle, with a centre c, could be drawn which would practically touch the four circles, and at the same time would pass through Huanillos.

III. The method of hyperbolas.—The method which I call that of hyperbolas is only another form of the method of circles. It is, however, useful in special cases, as, for instance, where we have the times of arrival of earthquakes at only two stations. Between Tokio and Yokohama, at which places I frequently obtain tolerably accurate time records, the method has been applied on several occasions with advantage. In the preceding example let us suppose that the only time records which we had were for Huanillos and Mejillones, and that the wave was felt at the latter place sixteen minutes or 960 seconds after it was experienced at the former. Calling these places h and m respectively, round m draw a circle equal to the 960 multiplied by the velocity with which the wave was propagated. It is then evident that the origin of this disturbance must be the centre of a circle which passes through h and touches the circle drawn round m. Join h m, cutting the circle round m in y. Bisect y h in v. It is evident that v is one possible origin for the disturbance. Next, from m, in the direction of h, draw any line m z p; join z h; bisect z h at right angles by the line o p n. Because ph = pz, it is evident that p is a second possible origin. Proceeding in this way a series of points lying to the right and left of v on the curve r v t may be found, and we may therefore say that the origin lies somewhere in the curve r v t. By increasing or decreasing our velocity we vary the position of the curve r v t, and, instead of a line on which our origin may be, we obtain a band. As the assumed velocity increases, the circle round m becomes larger, and has its limit when it passes through h, where the two arms of the curve r v t will close together and form a prolongation of the line m y h as the assumed velocity diminishes. The circle round m becomes smaller until it coincides with the point m. At such a moment the curve r v t opens out to form a straight line bisecting m h at right angles. The curve r v t is a hyperbola with a vertex v and foci h and m. Inasmuch as pm - ph = a constant quantity. If we have the time given at which the shock or wave arrived at a third station as at Iquique, it is evident that a second hyperbola r' v' t' might be drawn with Iquique and Huanillos as foci, and that the mutual intersection of these two hyperbolas with a third hyperbola, having for its foci Iquique and Mejillones, would give the origin of the wave. The obtaining of a mutual intersection would depend on the assumed velocity, and the accuracy of the result, like that of the method of circles, would depend upon the trials we made. The method here enunciated may be carried farther by describing hyperboloids instead of hyperbolas, the mutual intersection of which surfaces would, in the case of an earth wave, give the actual origin or centrum rather than the point above the origin or epicentrum.

IV. The method of co-ordinates.—Given the times at which a shock arrived at five or more places, the position of which we have marked upon a map, or chart, to determine the position on the map of the centre of the shock, its depth, and the velocity of propagation.

Commencing with the place which was last reached by the shock, call these places p, p₁, p₂, p₃, and p₄, and let the times taken to reach these places from the origin be respectively t, t₁, t₂, t₃, and t₄.

Through p draw rectangular co-ordinates, and with a scale measure the co-ordinates of p₁, p₂, p₃, and p₄, and let these respectively be a₁, b₁; a₂, b₂; a₃, b₃; a₄, b₄. Then if x, y, and z be the co-ordinates of the origin of the shock, d, d₁, d₂, d₃, and d₄, the respective distances of p, p₁, p₂, p₃, and p₄ from this origin, and v the velocity of the shock, we have

Because we know the actual times at which the waves arrived at the places p, p₁, p₂, p₃, p₄, we know the values t—t₁, t—t₂, t—t₃, t—t₄. Call these respectively m, p, q, and r. Suppose t known, then

Subtracting equation No. 1 from each of the equations 2, 3, 4, and 5, we obtain,

Now let v² = u, and 2v² t = w.

Then

We have here four simple equations containing the four unknown quantities x, y, u, and w.

x and y determine the origin of the shock. The absolute velocity v equals v u. From v and w we obtain t. Substituting x, y, v, and t in the first equation we obtain z.

We have here assumed that the points of observation have all about the same elevation above sea level.

If it is thought necessary to take these elevations into account, a sixth equation may be introduced.

If the propagation of the wave is considered as a horizontal one, as would be done when calculating the position of the epicentrum or point above the origin, by means of the times of arrival of a sea wave, the ordinate z of the first five equations would be omitted. Working in this way the resulting four equations, viz.

remained unchanged.

Applying this method to the same example as that used as illustration for the two previous methods, we obtain for the co-ordinates of Mejillones, Iquique, Cobija, Tocopilla, and Huanillos, measured in geographical miles, and the times in Iquique time at which the wave reached each, as given in the following table; ox and oy being, drawn through Mejillones.

	Co-ordinates		Time of arrival
	OX	OY	h.	m.
Mejillones	a or 0	b or 0	8	46	p. m.
Iquique	a1 or 150	b1 or 96	8	40	„
Cobija	a2 or 36	b2 or 14	8	38	„
Tocopilla	a3 or 66	b3 or 31	8	32	„
Huanillos	a4 or 102	b4 or 58	8	30	„

From this data we find the co-ordinates x and y of this origin to be 85·8 and 56·7; whilst the velocity of propagation = 45 feet per second.

Measuring these ordinates upon the map, we obtain a centre lying very near Long. 71° 5' W. and Lat. 21° 22' S., a position which is very near to that which has already been obtained by other methods.

If instead of Huanillos we substitute the ordinates and time of arrival of the sea wave for Pabalon de Pica, another point for the origin will be obtained lying farther out at sea. To obtain the best result, the method to be taken will evidently be, first to reject those places at which it seems likely that some mistake has been made with the time observations, and then with the remaining places to form as many equations as possible, and from these to obtain a mean value. This is a long and tedious process, and as the time observations of this particular earthquake are probably one and all more or less inaccurate, it is hardly worth while to follow the investigation farther.

In this example, as in the preceding ones, it will be observed that it has been sea waves that have been dealt with, rather than earth vibrations. It is evident, however, that these latter vibrations may be dealt with in a similar manner.

In these determinations it will also have been observed that it is assumed that the disturbance has radiated from a centre, and, therefore, approached the various stations in different directions. If we assume that we have three stations very near to each other as compared with their distances from the origin, so that we can assume that the wave fronts at the various stations were parallel, the determination of the direction in which the wave advanced appears to be much simplified. The determination of the direction in which a wave has passed across three stations was first given by Professor Haughton.

Haughton’s method.—Given, the time of an earthquake shock at three places, to determine its horizontal velocity and coseismal line.

The solution of this is contained in the formula

When a, b, and c are three stations at which a shock is observed at the times t₁, t₂, and t₃; a, b, and c are the distances between a, b, and c, and ? is the angle made by the coseismal lines x a x, y b y, and the line a b, which are assumed to be parallel.

This I applied in the case of the Iquique earthquake, but owing to the smallness of the angles between the three stations a, b, and c, the result was unsatisfactory. The problem ought to be restricted, first, to places which are a long distance away from a centre, and, secondly, to places which are not nearly in a straight line. This problem may be solved more readily by geometrical methods. Plot the three stations a, b, and c on a map, join the two stations between which there was the greatest difference in the time observation. Let these, for example, be a and c. Divide the line a c at point d, so that a d : d c as the interval between the shock felt at a and b is to the interval between the shock felt at b and c. The line b d will be parallel to the direction in which the wave advanced.

The difference in time of the arrival of two disturbances.—In the various calculations which have been made to determine an origin based on the assumption of a known or of a constant velocity, we have only dealt with a single wave, which may have been a disturbance in the earth or in the water. A factor which has not yet been employed in this investigation is the difference in time between the arrival of two disturbances; one propagated, for instance, through the earth, and the other, for example, through the ocean. The difference in the times of the arrival of two waves of this description is a quantity which is so often recorded that it is well not to pass it by unnoticed. To the waves mentioned we might also add sound waves, which so frequently accompany destructive earthquakes, and, in some localities, as, for instance, in Kameishi, in North Japan, are also commonly associated with earthquakes of but small intensity. It was by observing the difference in time between the shaking and the sound in different localities that Signor Abella was enabled to come to definite conclusions regarding the origin of the disturbances which affected the province of Neuva Viscoya in the Philippines, in 1881; the places where the interval of time was short, or the places where the two phenomena were almost simultaneous, being, in all probability, nearer to the origin than when the intervals were comparatively large. I myself applied the method with considerable success when seeking for the origin of the Iquique earthquake of 1877. The assumptions made in that particular instance were, first, that the velocity of the disturbance through the earth was known, and, secondly, that the velocity with which a sea wave was propagated was also known.

A method similar to the above was first suggested by Hopkins. It depended on the differences of velocity with which normal and transversal waves are propagated.[86]

Seebach’s method.—To determine the true velocity of an earthquake, the time of the first shock, and the depth of the centre.

Let the straight line m, m₁, m₂, m₃ represent the surface of the earth shaken by an earthquake. For small earthquakes, to consider the surface of the earth as a plane will not lead to serious errors.

If an earthquake originates at c, then to reach the surface at m it traverses a distance h in the time t. To reach the surface at m₁ it traverses a distance h + x₁ in a time t₂. If v equals the velocity of propagation,

Seebach now says that if we have given the position of m or epicentrum of the shock, and draw through it rectangular axes like m m₃ and m t₃, and lay down on m m₃ in miles the distances from M of the various stations which have been shaken, and in equal divisions for minutes lay down on m t₃ the differences of time at which m, m₁, m₂, &c. were shaken, then m₁ t₁, m₂ t₂, &c. are the co-ordinates of points on an hyperbola. The degree of exactness with which this hyperbola is in any given case constructed is a check upon the accuracy of the time observations and the position of the epicentrum. The apex of the hyperbola is the epicentrum.

The intersection of the asymptote with the ordinate axis is the time point of the first shock, which, because the scale for time and for space were taken as equal, gives the absolute position of the centrum. This intersection is shown by dotted lines. Knowing the position of the centrum, we can directly read from our diagram how far the disturbance has been propagated in a given time.

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